Seminar

Riemann surfaces, which we now think of abstractly as smooth algebraiccurves over the complex numbers, were described by Riemann as graphs ofmultivalued holomorphic functions-in other words, branched covers of theRiemann sphere $\P^1$. The Hurwitz spaces, varieties parametrizing theset of branched covers of $\P^1$ of given degree and genus, are stillcentral objects in the study of curves and their moduli. In this talk, we'lldescribe the geometry of Hurwitz spaces and their compactifications, leadingup to recent work of Anand Patel and others.